ERRATA:
COMPUTING CM POINTS ON SHIMURA CURVES ARISING FROM COCOMPACT ARITHMETIC TRIANGLE
GROUPS
JOHN VOIGHT
This note gives some errata for the article Computing CM points on Shimura curves arising from cocompact arithmetic triangle groups [2].
(1) Page 4, paragraph 3: It is claimed that each fieldF appearing in Takeuchi’s list has strict class number 1. This is not correct: these fields all have class number 1, but the fields Q(√
3), Q(√
6), Q(cosπ/12), and Q(cosπ/15), arising from the classes IV, V, IX, XV, XVI, and XVII, have strict class number 2. This mistake does not affect the results in any other section.
The fact that all fields have class number one is enough for the discussion of quadratic rings in§5; for the statements of descent in§6 concerning minimal fields of definition within the broader framework of triangle groups, see work of Clark and the author [1].
(2) The shading in Fig. 1 is incorrect: the two fundamental domains along the imaginary axis belowiare split into two whereas the others are not. (This error was spotted by Noam Elkies.) The corrected picture is shown on the next page.
References
[1] Pete L. Clark and John Voight,Algebraic curves uniformized by congruence subgroups of triangle groups, in preparation.
[2] John Voight,Computing CM points on Shimura curves arising from cocompact arithmetic triangle groups, Algorithmic number theory (ANTS VII, Berlin, 2006), eds. Florian Hess, Sebastian Pauli, Michael Pohst, Lecture Notes in Comp. Sci., vol. 4076, Springer, Berlin, 2006, 406–420.
Date: November 13, 2011.
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2 JOHN VOIGHT
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